3-Manifolds and Applications to Geometry

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Proposal: DMS-0083097PI: Walter D. NeumannDespite the venerable history of the algebraic geometry, many difficultproblems remain even in low dimensions. Methods of low dimensionaltopology have proved very helpful here, and part of this project concernsa continuation of the investigator's successful application of suchmethods to the study of singularities of complex surfaces and the topologyof plane affine curves. One issue to be addressed is explicit analyticdescription of singularities with given topology. The general question isnotoriously difficult, but the investigator and J. Wahl have recently madea series of conjectures on realising rationally Gorenstein singularitieswith rational homology sphere links as explicit quotients of completeintersections and are making significant progress towards theirresolution. 3-manifold topology is also being applied to the study of theglobal topology and analytic classification issues for two-variablepolynomials. These issues are of intrinsic interest as well as havingpotential to contribute to the resolution of the famous JacobianConjecture. In a rather different direction, the investigator will studyinvariants of 3-dimensional manifolds that relate to number theory andalgebra, namely the so-called character and eigenvalue varieties and theBloch invariant. Connections are emerging between these invariants thatwill inform both topology and algebra/number theory.Algebraic Geometry, which is essentially the study of the zero sets offamilies of polynomials, has always been an important area offundamental research, and is also important in such diverseapplications as control theory and communications. The project willapply topological methods to algebraic geometry, to obtain results onhow to realise and classify families of polynomials that lead toparticular topologies. The methods of 3-dimensional topology are ofparticular importance to this project, and there is also feedback fromalgebra and number-theory to topology, as well as some interactionwith theoretical physics and cosmology. The project also studiesnew insights that these connections bring to all these fields.